## Abstract and Applied Analysis

### Algebraic Structures Based on a Classifying Space of a Compact Lie Group

Dae-Woong Lee

#### Abstract

We analyze the algebraic structures based on a classifying space of a compact Lie group. We construct the connected graded free Lie algebra structure by considering the rationally nontrivial indecomposable and decomposable generators of homotopy groups and the cohomology cup products, and we show that the homomorphic image of homology generators can be expressed in terms of the Lie brackets in rational homology. By using the Milnor-Moore theorem, we also investigate the concrete primitive elements in the Pontrjagin algebra.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 508450, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393444210

Digital Object Identifier
doi:10.1155/2013/508450

Mathematical Reviews number (MathSciNet)
MR3126732

Zentralblatt MATH identifier
1292.55006

#### Citation

Lee, Dae-Woong. Algebraic Structures Based on a Classifying Space of a Compact Lie Group. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 508450, 7 pages. doi:10.1155/2013/508450. https://projecteuclid.org/euclid.aaa/1393444210

#### References

• J. D. Stasheff, “Homotopy associativity of $H$-spaces. I, II,” Transactions of the American Mathematical Society, vol. 108, pp. 275–312, 1963.
• M. Arkowitz, “Co-$H$-spaces,” in Handbook of Algebraic Topology, pp. 1143–1173, North-Holland, Amsterdam, The Netherlands, 1995.
• H. Scheerer, “On rationalized $H$- and co-$H$-spaces. With an appendix on decomposable $H$- and co-$H$-spaces,” Manuscripta Mathematica, vol. 51, no. 1-3, pp. 63–87, 1985.
• J. F. Adams, “An example in homotopy theory,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 53, pp. 922–923, 1957.
• B. I. Gray, “Spaces of the same $n$-type, for all $n$,” Topology, vol. 5, pp. 241–243, 1966.
• D.-W. Lee, “On the same $n$-type structure for the suspension of the Eilenberg-Mac Lane spaces,” Journal of Pure and Applied Algebra, vol. 214, no. 11, pp. 2027–2032, 2010.
• C. A. McGibbon and J. M. Møller, “On spaces with the same $n$-type for all $n$,” Topology, vol. 31, no. 1, pp. 177–201, 1992.
• C. A. McGibbon, “Self-maps of projective spaces,” Transactions of the American Mathematical Society, vol. 271, no. 1, pp. 325–346, 1982.
• M. A. Kutbi, J. Ahmad, N. Hussain, and M. Arshad, “Common fixed point results for mappings with rational expressions,” Abstract and Applied Analysis, vol. 2013, Article ID 549518, 11 pages, 2013.
• S. M. Abusalim and M. S. M. Noorani, “Fixed point and common fixed point theorems on ordered cone $b$-metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 815289, 7 pages, 2013.
• E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, NY, USA, 1966.
• G. W. Whitehead, Elements of Homotopy Theory, vol. 61 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1978.
• R. Bott and H. Samelson, “On the Pontryagin product in spaces of paths,” Commentarii Mathematici Helvetici, vol. 27, pp. 320–337, 1953.
• P. J. Hilton, “On the homotopy groups of the union of spheres,” Journal of the London Mathematical Society, vol. 30, pp. 154–172, 1955.
• M. Arkowitz and C. R. Curjel, “Homotopy commutators of finite order. I,” The Quarterly Journal of Mathematics, vol. 14, pp. 213–219, 1963.
• M. Arkowitz, “Commutators and cup products,” Illinois Journal of Mathematics, vol. 8, pp. 571–581, 1964.
• F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, “Torsion in homotopy groups,” Annals of Mathematics, vol. 109, no. 1, pp. 121–168, 1979.
• Y. Félix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, vol. 205 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2001.
• J. W. Milnor and J. C. Moore, “On the structure of Hopf algebras,” Annals of Mathematics, vol. 81, pp. 211–264, 1965.
• K. Morisugi, “Projective elements in $K$-theory and self maps of ${\Sigma \mathbb{C}P}^{\infty }$,” Journal of Mathematics of Kyoto University, vol. 38, no. 1, pp. 151–165, 1998.